Fermat cofactors

[ewmayer]

[By way of cross-referencing]

I just posted details/results for my tests of the F25-F30 cofactors here. Executive Summary:

o My Fermat-PRP Res64 values (starting with my saved Pepin-test residues and doing one more mod-squaring) match Yar's, but we need George to tweak his code to also print the Res64 for the ensuing Suyama cofactor-PRP postprocessing step;

o Once we have the above added-Res64-print in place, it would be nice if someone could run Fermat-PRP/Suyama for F27 in order to cross-compare that one. Andreas Höglund did a PRP-CF run of F27 already some years ago, but George's code-at-the-time (2009) which Andreas used was doing a direct-PRP-test of the cofactor, whose results cannot be cross-checked against the Pepin+Suyama one.

[ATH]

Amazing work Ernst on those cofactor tests specifically the latter ones. You had the patience for those multiyear tests, as well as all the work on MLucas to make it happen.

I assume the "Fermat-PRP/Suyama" test of F27 is a type 5 PRP-CF test? I might run it if no one else does it, but not right now or the next few weeks. Maybe it is better to wait anyway in case George feels like and have time for this "tweak". It sounds from your description like it is a minor change/addition.

[ET_]

Thank you Ernst. It looks that the original page from Prof. Keller will be updated, once we have all the checks done.

Please keep us informed!

[ewmayer]

Originally Posted by ATH

Amazing work Ernst on those cofactor tests specifically the latter ones. You had the patience for those multiyear tests, as well as all the work on MLucas to make it happen.

I assume the "Fermat-PRP/Suyama" test of F27 is a type 5 PRP-CF test? I might run it if no one else does it, but not right now or the next few weeks. Maybe it is better to wait anyway in case George feels like and have time for this "tweak". It sounds from your description like it is a minor change/addition.

Thanks, Andreas (and Luigi).

Yes, 'type 5' is the GIMPS jargon - same as type 1, i.e. a Fermat-PRP, a^(N-1) (mod N), but followed by the Suyama-style cofactor-PRP step. The checksum-for-Suyama-result is a more or less trival code-modification, but crucial for cross-validation of cofactor-PRP results. I sent George (and AaronB and JamesH) email a few weeks ago suggesting that the JSON result format for PRP-CF be modified from simply noting 'C' or 'P' for the cofactor to including the Suyama-step Res64 for the 'C' cases, i.e. the vast majority. The preceding much-longer Fermat-PRP test of N could be reported separately, same as a regular PRP, and allowing all the same checks, i.e. Gerbicz and even proof/cert.

(It would be really cool to also support upload of Fermat-PRP residues to the server, because that would allow really fast server-side auto-running of the cofactor-PRP step for N where future deeper factoring work turns up a factor.)